Research on the initiation pressure criterion of directional hydraulic fracturing in coal mine

Abstract

Directional hydraulic fracturing (DHF) is more and more widely used in coal mines in China for hard roof and coal burst control. The key to this technology is to determine the crack initiation pressure that affected by the shape of the artificial notch and the stress state around the fracturing hole. Reasonable and simple formula for fracturing pressure calculation is essential since the fracturing pump used in coal mines is usually limited by the harsh conditions and hardly replaced once selected. Based on the superposition principle, the simplified 2D model of DHF was established as the elliptical hole with the internal pressure and solved by using the complex functions method. The analytical solution of tangential stress on the inner surface was obtained meanwhile the corresponding criterion of fracturing pressure can be set up. Considering the characteristics of DHF in coal mines, we further got a simplified formula that controlled by the ratio of major to minor axis of the ellipse-like notch, the ratio of the minimum to the maximum principal stress, as well as the tensile strength of the rock. The formula also gave a guide to the design of the notch that major diameter should be at least twice the minor diameter, and the optimal solution for the ratio is to 2～4 and recommended 4, which can resist the initiation pressure to a large extent affected by the in-situ stress. Once the pressure of the fracturing fluid is high enough to satisfy the equation cracks would arise at the tips of the notch along the major axis which belongs to mode Ⅰ crack and would grow unsteadily and rapidly. A PFC simulation model was used to verify the analysis, the results of which are very consistent with the theoretical solutions.

1. Introduction

With the increase of the mining depth and mining intensity, the coal mining industry is entering an era of confronting with severe dynamic disasters due to the high static stress and severe perturbance stress [1]. Coal burst is a dynamic disaster induced by elastic strain energy emitted in a sudden, rapid and violent way from coal or rock masses, often accompanied by an air-blast or windblast and violent failures which can disrupt mine ventilation, pose a threat to miners due to flying matters, and may also cause a massive release of coal seam methane and spread explosive dust into the air [2,3]. Most mining countries including China, Poland, USA, Australia, South Africa are suffering from coal burst disaster [4]. According to statistics, there are approximately 138 burst-prone coal mines in China as of 2021 and the number keeps increasing [5]. Researches show that the majority of the coal burst disasters are controlled and induced by the adjacent strong massive roof strata [6]. Thus, the presence of strong massive roof strata close to the coal seam is taken as an indicator of coal burst hazard in practice. Weakening the strong roof strata is an effective way of mitigating coal burst risk during coal mining and is the common practice in current application [7]. Two common techniques used for weakening strata are roof blasting and water injection. As for roof blasting, the implementation is complicated and has lots of potential security risks therefor is increasingly restricted to be used in China. When it comes to water injection, its weakening effect is limited in rock strata as the softening coefficient of the strong roof strata is normally very poor [8]. To realize the goal of safe and high-efficiency mining, hydraulic fracturing (HF) technique has been extensively applied in coal mines, especially in enhancing the permeability of coal seams, improving coal seam methane extraction, preventing gas outburst, enhancing the caveability of top coal, destressing stress-concentrated coal seam, etc., which demonstrates great prospect of application [[9], [10], [11], [12], [13], [14], [15], [16]]. However, as for normal nondirectional hydraulic fracturing, the water pressure and flow rate required to fracture the strong massive roof strata is very high and the control of crack propagation is not easy [17]. Thus, the fracturing effect of nondirectional hydraulic fracturing is not remarkable [18]. In contrast, these problems can be solved by directional hydraulic fracturing (DHF). Directional hydraulic fracturing refers to cutting an artificial notch at the predesigned direction with a specifically-designed cutter, and injecting high-pressure water to fracture the rock mass [[19], [20], [21]]. As directional hydraulic fracturing can slice rock strata at the designed direction thereby thinning rock layers and lowering rock strength, it is becoming one of the main directions of development in coal burst control [22].

Numerous progresses of hydraulic fracturing have been achieved in theoretical, industrial and laboratory research especially with the advanced computer software are gradually put into use [[23], [24], [25]]. Hydraulic fracturing was applied in more fields, such as gas reservoir management, repeated fracturing, sandstone treatment, geothermal resources exploitation, nuclear waste storage, coal mine disaster control etc. In the exploitation of petroleum, natural gas and shale gas, the main method to control the directional initiation of hydraulic fracturing is the perforation operation, and the high-pressure water jet and abrasive jet are gradually evolved from the perforation operation [26]. In coal mining engineering, Central Mining Institute of Poland first applied directional hydraulic fracturing technology to the treatment of strong roof strata. In recent years, the directional hydraulic fracturing technology has been studied and applied in some coal mines in China [27]. From literature statistics, most achievements of hydraulic fracturing are concentrated in the field of oil and gas exploitation, and researchers mainly focus on increasing the number and distance of cracks that is improving the rock permeability [28,29], in which the key issues include the propagation of fracturing cracks in heterogeneous rock mass and their interaction with nature fractures [[30], [31], [32], [33], [34]], while the pressure required to initiate the crack and its changes during hydraulic fracturing process are becoming less important [35]. Compared with the field of oil and gas exploration, hydraulic fracturing in coal mines especially for hard roof and coal burst control has its own characteristics that the initiation pressure and controlling of crack propagation direction are prioritized since the limited of the space in roadways, thus the directional hydraulic fracturing is by cutting an orientated notch is getting much more popular [[36], [37], [38], [39], [40], [41]]. In China, this technique was tried in several coal mines and some successfully fractured rock strata while the others not since the unreasonable pressure parameter [42]. Although several theoretical or numerical models of DHF have been established to calculate the initiation pressure [43,44], systematic fundamental investigation is required to improve its applicability. Hence, this paper will focus on the crack initiation mechanism of directional hydraulic fracturing under mining stress condition, by establishing a simplified two-dimension model and analyzing the influencing factors of the ignition pressure to provide a guide for the parameter and perforation cutter design. The research is of scientific importance in revealing the directional hydraulic fracturing mechanism.

2. Mechanism and criterion of directional hydraulic fracturing (DHF)

2.1. Mechanical model of DHF

The DHF is similar to the perforation technique used in oil and gas industries, which firstly prefabricate an artificial notch usually with a special drill around the borehole, as shown in Fig. 1. The essence of DHF is the generation of a spatially oriented notch in the rock mass and after high pressure liquid injected into the borehole, the cracks would propagate from the tips of the oriented notch thereby divide the rock layers into blocks or plates with determined sizes and forms. Such a process is owing to the generation of the so-called artificial notch with exactly spatial orientation in the borehole surroundings. Fig. 2 shows the photos in laboratory experiments carried out by Poland experts and field application in a colliery accomplished by the authors using drilling speculum [17]. This artificial notch delimits the direction of fracture propagation and its rise is induced by the high-pressure liquid. Both the integrity and strength of hard roof are weakened after fractured, as a result the sudden roof falling with large area is avoided and the coal burst danger reduced ultimately.

Fig. 1.

Schematic diagram of DHF.

Fig. 2.

Photographs of the profile of the artificial notch in lab (Panel (a)) and on-site engineering (Panel (b)).

As the directional notch has a certain spatial shape, the distribution characteristics of the surrounding stress after the injection of high-pressure water can be analyzed using the “plate with holes” solutions of elasticity theory to obtain the stress distribution around the notch and crack initiation criterion. Fig. 3 shows a mechanical model of the DHF, which can be simplified to a two-dimensional plane strain problem for processing. Where ${\sigma }_1$ and ${\sigma }_3$ are the maximum and minimum principle stress, respectively, $a$ is the semi-major axis and $b$ is semi-minor axis of the ellipse, $p$ is the pressure of the injected liquid. According to the superposition principle, the mechanical model shown in Fig. 3 can be decomposed of three parts, as shown in Fig. 4, that is an elliptic hole subjected to uniform internal pressure and an elliptic hole subjected to different uniaxial stresses. It can be solved using the complex variables function of elasticity.

Fig. 3.

The mechanical model of a directional notch in DHF.

Fig. 4.

Three components of the mechanical model that component 1 under the maximum principal stress (Panel (a)), component 2 under the minimum principal stress (Panel (b)) and component 3 under the fracturing fluid pressure (Panel (c)).

2.2. Solution of the model

Based on the complex analysis methods of infinite plate with an elliptic hole, first covert the cartesian coordinate system to curved coordinate using conformal transformation. Map the perimeter of the ellipse denoted as $L$ on the z-plane to a unit circle denoted as $\gamma$ on the $\zeta$ plane, and the infinite field $D$ outside the elliptical hole to the inner region $G$ of the unit circle, as shown in Fig. 5. And as the $\alpha$ is positive turning counterclockwise whereas $\theta$ in Fig. 5(b) is on the contrary, so the points of $A$、 $B$、 $C$、 $D$ in Fig. 5 (a) correspond to $A^{\prime }$、 $B^{\prime }$、 $C^{\prime }$、 $D^{\prime }$ in Fig. 5 (b), in addition the origin point in Fig. 5(b) that $\zeta =0$ corresponds to the points infinitely far from the ellipse in Fig. 5(a).

Fig. 5.

Conformal mapping of the coordinates system.

The following mapping function was adopted as Eq. (1).

$$z=W\left(\zeta \right)=R\left(\frac{1}{\xi }+m\xi \right)$$ (1)

where, $R=\frac{a+b}{2}$， $m=\frac{a-b}{a+b}$，both are real numbers, determined by the major and minor axis of the ellipse, and $\zeta =\xi +\mathrm{i}\eta$.

For the case of Fig. 4(a), set the maximum principal stress ${\sigma }_1$ as $q$, the boundary conditions are shown as Eq. (2).

$${\sigma }_1=q,{\sigma }_2=0,{\sigma }_3=0,\bar{X}=\bar{Y}=0,X=Y=0$$ (2)

where, $\bar{X}$ and $\bar{Y}$ are the surface force, while $X$ and $Y$ are the body force. Thus, one can obtain the complex function expression of the stress component as Eq. (3).

$$\{\begin{array}{l}{\sigma }_{\theta }+{\sigma }_{\rho }=q\mathrm{R}\mathrm{e}\frac{\left(2{\mathrm{e}}^{2i\alpha }-m\right){\zeta }^2-1}{m{\zeta }^2-1}\\ {\sigma }_{\theta }-{\sigma }_{\rho }+{\tau }_{\rho \theta }=\frac{q\left(m{\rho }^4+{\zeta }^2\right){\zeta }^2}{{\rho }^4\left(m-\frac{{\zeta }^2}{{\rho }^4}\right)\left(m{\zeta }^2-1\right)}\left[2{\mathrm{e}}^{2i\alpha }-m+m\frac{1+m{\zeta }^2-2{\mathrm{e}}^{2i\alpha }{\zeta }^2}{m{\zeta }^2-1}\right]+\\ \frac{q}{{\rho }^2\left(m-\frac{{\zeta }^2}{{\rho }^4}\right)}\left[{\mathrm{e}}^{-2i\alpha }-\frac{3{\mathrm{e}}^{2i\alpha }{\zeta }^2+m{\mathrm{e}}^{2i\alpha }-m^2-1}{m{\zeta }^2-1}{\zeta }^2+\frac{{\mathrm{e}}^{2i\alpha }{\zeta }^2+m{\mathrm{e}}^{2i\alpha }-m^2-1}{{\left(m{\zeta }^2-1\right)}^2}2m{\zeta }^4\right]\\ \end{array}$$ (3)

Noting that $\zeta =\rho \left(\cos \theta +\mathrm{i}\sin \theta \right)$, from Eq. (3) we can obtain the expressions of ${\sigma }_{\theta }$, ${\sigma }_{\rho }$ and ${\tau }_{\theta \rho }$ by separating the real and imaginary parts. As the hole and cavity problem, what we concerned most important is the stress on the inner wall (boundary) of the hole where ${\sigma }_{\rho }=0$, $\rho =1$, define points on the boundary as $\zeta =\sigma =\rho {\mathrm{e}}^{\mathrm{i}\theta }={\mathrm{e}}^{\mathrm{i}\theta }$, so we can have the tangential stress expressed of case in Fig. 4(a) as Eq. (4).

$${\sigma }_{\theta }=q\frac{1-m^2+2m\cos 2\alpha -2\cos 2\left(\theta +\alpha \right)}{1+m^2-2m\cos 2\theta }$$ (4)

and as for the case of Fig. 4(b) the process and solution are the same only need to rotate the coordinate axis. Let the angle between the stress component and the major axis of the ellipse are $\alpha ,\alpha +\frac{\pi }{2}$, for Fig. 4(a) and (b), and we can get the superimposed results of the tangential stress on the hole wall shown as Eq. (5).

$${\sigma }_{\theta }=q\frac{1-m^2+2m\cos 2\alpha -2\cos 2\left(\theta +\alpha \right)}{1+m^2-2m\cos 2\theta }+\lambda q\frac{1-m^2+2m\cos 2\left(\alpha +\frac{\pi }{2}\right)-2\cos 2\left(\theta +\alpha +\frac{\pi }{2}\right)}{1+m^2-2m\cos 2\theta }$$ (5)

For the situation of Fig. 4(c) that subjected to a uniform liquid pressure, we can obtain the surface force components as Eq. (6).

$$\{\begin{array}{l}\bar{X}=q\cos \left(N,x\right)=pL\\ \bar{Y}=q\cos \left(N,y\right)=pm\\ \bar{X}+\mathrm{i}\bar{Y}=p\left(L+\mathrm{i}m\right)\end{array}$$ (6)

and hence we write the expression as Eq. (7):

$$\left(\bar{X}+\mathrm{i}\bar{Y}\right)ds=p\left(Lds+\mathrm{i}mds\right)=p\left(dy-\mathrm{i}dx\right)=-\mathrm{i}p\left(dx+\mathrm{i}dy\right)=-\mathrm{i}pdz$$ (7)

The surface force on the inner hole surface is a balance system, the resultant force equals to zero and at the same time as well as the stress outside that is ${\sigma }_x\left(\infty \right)={\sigma }_y\left(\infty \right)=\tau \left(\infty \right)=0$, thereby on the boundary we have $f\left(\sigma \right)=\mathrm{i}\int \left(F_x+\mathrm{i}{F}_y\right)ds=p\int dz=pz$, substituted into Eq. (1) to obtain Eq. (8).

$$f\left(\sigma \right)=\mathit{pR}\left(\frac{1}{\zeta }+m\zeta \right){|}_{\zeta =\sigma }=\mathit{pR}\left(\frac{1}{\sigma }+m\sigma \right)$$ (8)

Based on the classical solution of elasticity, we can obtain the expressions of ${\sigma }_{\theta }$、 ${\sigma }_{\rho }$、 ${\tau }_{\theta \rho }$ for Fig. 4(c). Also, the most important stress distribution we cared is on the inner boundary that $\rho =1$ as well as ${\sigma }_{\rho }=p$, therefore, we obtain Eq. (9).

$$\{\begin{array}{l}{\sigma }_{\theta }=-p\frac{1-3m^2+2m\cos 2\theta }{1+m^2-2m\cos 2\theta }\\ {\sigma }_{\rho }=p\end{array}$$ (9)

Under the hydraulic pressure, it is easily to generate a tensile stress around the hole and the maximum tensile stress appears at $\theta =0;\pi$ that is ${\sigma }_{\theta }{|}_{\theta =0;\pi }=-p\frac{1+3m}{1-m}$.

By superimposing Eq. (5) and Eq. (6) we can obtain the stress expression on the inner wall of the elliptical hole in Fig. 2. And for hydraulic fracturing, the rock breakage is caused by the tangential stress ${\sigma }_{\theta }$, shown as Eq. (10).

$${\sigma }_{\theta }=q\frac{1-m^2+2m\cos 2\alpha -2\cos 2\left(\theta +\alpha \right)}{1+m^2-2m\cos 2\theta }+\lambda q\frac{1-m^2+2m\cos 2\left(\alpha +\frac{\pi }{2}\right)-2\cos 2\left(\theta +\alpha +\frac{\pi }{2}\right)}{1+m^2-2m\cos 2\theta }-p\frac{1-3m^2+2m\cos 2\theta }{1+m^2-2m\cos 2\theta }$$ (10)

where, $q$ is the value of the maximum principal stress, $\lambda$ is the ratio of maximum principal stress to minimum principal stress, $\alpha$ is the angle between the major axis of the elliptical hole and the maximum principal stress, $\theta$ is eccentric angle of any point on the ellipse, and $q$ is the pressure of the liquid injected into the hole.

The pressure of the liquid must high enough to generate negative tangential stress and exceed the tensile strength of the rock so that hydraulic fracturing can be processed. As the special conditions and requirements such as narrow roadway space, explosion-proof in underground coal mines, the determination of the reasonable pressure of hydraulic fracturing is essential since temporary change of bumps is not easy or costs too much.

We can establish a criterion of crack generation as Eq. (11) based on Eq. (10).

$$\{\begin{array}{l}\left|{\sigma }_{\theta }\right|>R_t\\ \Rightarrow q\frac{1-m^2+2m\cos 2\alpha -2\cos 2\left(\theta +\alpha \right)}{1+m^2-2m\cos 2\theta }+\lambda q\frac{1-m^2+2m\cos 2\left(\alpha +\frac{\pi }{2}\right)-2\cos 2\left(\theta +\alpha +\frac{\pi }{2}\right)}{1+m^2-2m\cos 2\theta }\\ -p\frac{1-3m^2+2m\cos 2\theta }{1+m^2-2m\cos 2\theta }<-R_t\end{array}$$ (11)

Eq. (10) provides the capability of a pump needed and also a guide of optimization of the notch design. As the notch shape and the knowledge of the stress field are determined, we can calculate the liquid pressure for the initiation of the crack at the tip of the notch.

As an example, let $a=b$, ${\sigma }_1=q$, ${\sigma }_3=\lambda q$, $\alpha =0$, based on Eq. (11) the maximum tensile stress appears on the tip of the ellipse, that means $\theta =0$, under this condition, the elliptical hole becomes the most common circle hole, and substitute these into Eq. (11), we obtain $p=3{\sigma }_3-{\sigma }_1+R_t$, which is the classic formula of hydraulic fracturing in the circle borehole, which also proved the correctness of the formula.

2.3. Critical injection stress of directional hydraulic fracturing in coal mine

Normally the parameters of the device used to cut the artificial notch in the borehole are determined, for instance artificial notch made by mechanical cutting device shown as Fig. 2, the ratio of the diameter and the height, also the major and minor axes of the elliptical hole, is 2–4. Unfortunately, the stress condition around the roadways is hardly measured in the underground coal mine, therefore the worst case should be taken into consideration for the critical stress calculation that tangential stress around the notch is compressive at any point around the notch before the fracturing liquid injected, and the maximum compressive stress generated when $\alpha ={90}^{\circ }$ as well as $\theta =0;\pi$, therefore Eq. (5) can be simplified as Eq. (12).

$${\left({\sigma }_{c\max }\right)}_{\theta =0,\pi ;\alpha =\frac{\pi }{2}}={\sigma }_1\frac{3+m}{1-m}-{\sigma }_3=\left(1+\frac{2a}{b}-\lambda \right){\sigma }_1$$ (12)

After the fracturing liquid was injected into the hole the maximum tensile stress also appeared at where $\theta =0$ and $\theta =\pi$ as in Eq. (23).

$$\sigma {}_{t\max }={\left({\sigma }_{\theta }\right)}_{\theta =0,\pi }=-p\frac{1+3m}{1-m}=-p\left(2\frac{a}{b}-1\right)$$ (13)

so, the critical injection stress of directional hydraulic fracturing in coal mine can be estimated using Eq. (14).

$$p\ge \frac{1}{2k-1}\left(\left(1+2k-\lambda \right){\sigma }_1+R_t\right)$$ (14)

where, $k=\frac{a}{b}$, $k\ge 1$ and $\lambda =\frac{{\sigma }_3}{{\sigma }_1}$, $0<\lambda \le 1$, $R_t$ is the tensile strength of the rock mass.

It can be seen that the pressure is mainly determined by the shape of the notch and the in-situ stress around the hydraulic fracturing sites as shown in Fig. 6 and Fig. 7, for instance ${\sigma }_1=20$ MPa, $R_t=5$ MPa, respectively, and Fig. 6 shows a consistent tendency in different stress field. The hydraulic pressure required in the circle hole ($k=1$) is 65 MPa while 35 MPa in the elliptical hole with $k=2$ and about 26.43 MPa when $k=4$, decreases nearly 53.8% and 40.7%, respectively, however only reduces 11.0% from $k=4$ to $k=6$. It also gives a guide to the design of the notch-cutting device that major diameter should be at least twice the minor diameter. The optimal solution for the ratio $k$ is to 2～4 and recommended 4, since no significantly reduce of the pressure with the increase of $k$ while on the contrary the cutting device would easily break in the hole. Fig. 7 shows that a negative linear relationship between the critical fluid pressure and $\lambda$ which means a uniform in-situ stress condition facilitates the crack initiation. But the slopes are quite different between $k=1$ and $k>1$ which demonstrates hydraulic fracturing in the circular hole significantly controlled by the in-situ stress, assuming without the effect of heterogeneity and natural fracture. The decline rate of the critical pressure with the increase of $\lambda$ is similar when $k\ge 2$, much smaller than $k=1$. The larger of the ratio $k$, the smaller changes of critical pressure, which means reasonable shape of the artificial notch could offset the crack initiation pressure and even propagation modes affected by the in-situ stress.

Fig. 6.

Pressure of fracturing fluid with the shape of the notch (k) under different stress state.

Fig. 7.

Pressure of fracturing fluid in different stress state with different shape of the notch (k).

Eq. (14) is not only a criterion of the initiation pressure but also a condition for directional propagation of the crack although lots of simplifications and assumptions were adopted. As long as the pressure of injection fluid is high enough to satisfy Eq. (14) a crack would arise at the tips of the notch along the major axis and be vertical to the direction of fluid pressure that is minimum principal stress on the inner surface of the hole. Cracks on the tip the notch belong to mode Ⅰ crack and will grow unsteadily and rapidly than other cracks at different direction which however would stop sooner. Eq. (14) is very convenient for mining engineers to calculate the pressure of the fracturing pump in the case of enough information of the stress field around the hydraulic fracturing sites, which sometimes need to be estimated based on reported research and the errors is tolerable.

3. Criterion verification of DHF

3.1. PFC numerical simulation model

The commercial particle flow code software PFC2D developed by Itasca is used to simulate the crack initiation and propagation of DHF, as shown in Fig. 8. The selection of the meso-mechanical parameters of the rock mass is the key for simulation. Based on the principle and published methods for parameter calibration, the linear parallel bond model with the uniform distribution of particles is adopted. The minimum radius of the particles is 0.018 m and the ratio of maximum radius to the minimum is 1.66, the porosity is 0.99 at model reached equilibrium.

Fig. 8.

The stress - strain curve (Panel (a)) and image (Panel (b)) of uniaxial compression test using the calibrated micro-parameters.

The macro-mechanical properties of the rock block come from a colliery in the city of Xuzhou, Jiangsu province, China. Table 1 shows the laboratory results using MTS C64.106/10. Two calibration models of uniaxial compression test and Brazilian splitting tensile test were set to correct the micro-parameters used in the simulation model later. After sever debugging, reasonable micro-parameters were obtained as shown in Table 2 and corresponding macro-curves of strain-stress were shown as Figs. 8 and 9.

Table 1.

Macro-mechanical parameters of rock mass.

Lithology	Density/kg/m3	Uniaxial compressive strength/MPa	Tensile strength/MPa	Elastic Modulus/GPa	Poisson's ratio	Cohesion/MPa	Friction/°
Medium sandstone	2550	57.28	6.48	14.45	0.23	4.50	25°34＇

Table 2.

Micro-mechanical parameters used in the simulation model after calibration.

Micro-mechanical parameters	Value
Minimum particle radius/m	0.018
Ratio of maximum radius to the minimum radius	1.66
Particle density kg/m3	2550
Porosity	0.09
Particle friction coefficient	0.50
Normal bond strength/MPa	16.5
Tangential bond strength/MPa	26.0
Bond stiffness ratio	2.50
Adhesion modulus/GPa	8.0

Fig. 9.

The stress - strain curve (Panel (a)) and image (Panel (b)) of tensile strength test using the calibrated micro-parameters.

3.2. Verification of DHF initial stress

The sizes of the two simulation models are both 4 m × 4 m with an elliptical notch in the middle as shown in Fig. 10. The major axes of both notches are vertical to the maximum principal stress, while the ratio of the major and minor axes is selected as 4. The maximum principal stress ${\sigma }_1$ applied on the model equals to 12 MPa as well as ${\sigma }_3$ equals to 8 MPa. To ensure the success fracturing, the initial pressure of injection water is calculated by hydraulic fracturing formula in the circle borehole which equals to 18.48 MPa, taken a safety factor of 1.5 into consideration, the fracturing pressure of 28 MPa is adopted in this model.

Fig. 10.

Simulation models of DHF with elliptical notch.

The fluid pressure during the hydraulic fracturing process is monitored until no crack generated which lasted 400 s in total as shown in Fig. 11 that is as typical hydraulic fracturing curve. The fluid pressure rises to the maximum of 15.5 MPa and dropped rapidly which demonstrated cracks had been generated. From the curve combined with the numbers cracked bonds can determine the initiation pressure in this case is 15.5 MPa very close to the theoretical value calculated by Eq. (14) that is about 15.21 MPa, both are smaller than that needed in a circular borehole. Fig. 12 shows the diameter of the induced main crack along the major axis and numbers of bond rupture during DHF, both can be divided into 3 stage, similar like the change of the fluid pressure. The diameter and rupture number show a consistent tendency that means few bifurcations generated which facilitates the crack directional growth and propagation. Once the initial crack generated the fracturing fastened and extended to 85% of the total length within 56% of the total time, and then the fracturing turns into a stable propagation takes up 45% of the total time and stops at last.

Fig. 11.

Fluid pressure changes during the DHF simulation.

Fig. 12.

Crack diameter and bond rupture during DHF.

3.3. Field test and verification

Several on-site engineering projects have been conducted among which the typical case is at Jining NO.3 coal mine in Shandong Energy Group Co., Ltd. The detailed geological conditions of the on-site implementation and some related research results can be found in reference [17]. And some necessary information that used to calculate the fracture pressure are as follows, the buried depth of the longwall mining workface is 660 m, the target stratum for fracturing is a 28 m thick medium sandstone with a tensile strength of 8～10 MPa on average. The in-situ stress was measured by stress relieving method and the maximum and minimum principal stress were 20.52 MPa and 16.14 MPa, respectively. Based on Eq. (14) we can calculate the pressure required to fracture the sandstone were 25.22 MPa–25.50 MPa when $k=4$ and $R_t=8\sim 10$ MPa. The pressure variation trend during directional fracturing was shown in Fig. 13 and the data in this figure were obtained once every 10 s using the digital and direct explicit pressure gauges. It can be clearly seen that the maximum pressures are 18.0 MPa, 22.0 MPa and 24.0 MPa at three different fracturing points that all very closed to the calculated values, meanwhile they are all slightly smaller than the theoretical value. Many factors can affect the initiation pressure during field tests including the in-situ stress state, the strength of the rock mass, the nature fissures as well as the quality of the sealing and the shape of the artificial notch, among which the in-situ stress and the fabric of the stratum may play a dominant role not only in the initial pressure but also the crack development. In this case, the main reason that the initiation pressure is less than the calculated value should be attributed to the actual tensile strength of the stratum is lower than the laboratory test value. The above case indicates that the stress value obtained by the elliptical model are relatively close to the actual monitoring results and can be used for engineering design and calculation and a certain safety factor is recommend if there is not enough accurate data.

Fig. 13.

Curves of pressure variation trend during directional fracturing.

4. Conclusions

Determining of the initiation pressure and reasonable shape of the artificial notch are crucial to the success of the directional hydraulic fracturing (DHF) in coal mines, both of which need to obtain the stress distribution around the fracturing borehole. As the shape of the notch generated in the borehole is like an ellipse, we established a simplified two-dimension model and analyzed the problem using the complex functions method of elasticity theory. The DHF model can be decomposed into three components based on superposition principle. The expression of tangential stress on the inner surface of the elliptical hole was obtained, therefor the criterion of the initiation pressures can be set up if the tensile tangential stress appeared and the value exceeded the tensile strength of the rock. The general formula of the critical pressure is controlled by in-situ stress state as well as the shape and orientation of the notch, and different on different position, which hardly used by coal mine engineers.

Considering the characteristics of DHF in coal mines, we further got a simplified formula that controlled by the ratio of major to minor axis of the artificial notch, the ratio of the minimum to the maximum principal stress, as well as the tensile strength of the rock. The formula also gave a guide to the design of the notch that major diameter should be at least twice the minor diameter, and the optimal solution for the ratio is to 2～4 and recommended 4, which can resist the initiation pressure to a large extent affected by the in-situ stress. Once the pressure of the fracturing fluid is high enough to satisfy the equation in the paper a crack would arise at the tips of the notch along the major axis which belongs to mode Ⅰ crack and would grow unsteadily and rapidly. Numerical simulation implemented by PFC2D verified the correctness of the formula. The diameter and rupture number show a consistent tendency that means few bifurcations generated which facilitates the crack directional growth and propagation. The first unstable propagation stage after cracking initiated contributed 85% of the maximum diameter within 45% total fracturing time.

Author contribution statement

Hu He: Conceived and designed the experiments; Performed the experiments; Wrote the paper.

Ruyi Cheng; Junming Zhao; Zhengbing Men: Analyzed and interpreted the data.

Zonglong Mu: Contributed reagents, materials, analysis tools or data.

Data availability statement

Data will be made available on request.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.